 \documentclass[12pt]{article}
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\graphicspath{{./Graphs/}}
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\newtheorem{definition}{Definition}
\newtheorem{claim}{Claim}
\newtheorem{example}{Example}
\newtheorem{proposition}{Proposition}
\newtheorem{lemma}{Lemma}
\newtheorem{remark}{Remark}
\newtheorem{corollary}{Corollary}
\newtheorem{implication}{Implication}
\newtheorem{assumption}{Assumption}
\newtheorem{property}{Property}
\newcommand{\bit}{\begin{itemize}}


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\usepackage{graphicx}
\usepackage{tikz}
\graphicspath{ {./Graphs/} }
\begin {document} 

\begin{figure}

\begin{center}
\begin{tikzpicture}

\begin{axis}[
    axis y line=middle, 
    axis x line=middle,
    xlabel = $x_1$,
    ylabel = {$U^V_1$},
    yticklabels={,,,,,,},
    xticklabels={,,,,},
     legend pos={south west}
]
\addplot [
 domain=-0.5:0.5, 
    samples=100, 
    color=black,
    style=ultra thick,
]
{-(x-1)^2  + 1};

 \node [ above right] at (axis cs:  0.25,0) {$x'$};\node [ above left] at (axis cs:  -0.25,0) {$-x'$};
 
\addplot[color=black,mark=*,mark options={fill=white}]
   coordinates {
      
        (0.25,-1+1)}; 
        \addplot[color=black,mark=*,mark options={fill=white}]
   coordinates {
      
        (-0.25,-1 +1)}; 
\addplot [
    domain=-0.5:0.5, 
    samples=100, 
    color=gray,
    ]
    {-(x+1)^2 + 1};

\addplot [name path=a][
 domain=-0.5:0.5,
    samples=100, 
    color=gray,
    style=dashed
]
{-(x+1)^2+0.5 +1};
\addplot [name path=b][
 domain=-0.5:0.5,
    samples=100, 
    color=gray,
    style=dashed
]
{-(x+1)^2-0.5 +1}; 

 \addplot [name path=c][
     domain=-0.5:0.5, 
    samples=100, 
color=black,
    style=dashed,ultra thick,
    ]
{-(x-1)^2+0.5 +1};
\addplot [name path=d][
 domain=-0.5:0.5,
    samples=100, 
    color=black,
    style=dashed,ultra thick,
]
{-(x-1)^2-0.5 +1}; 


\node [below] at (axis cs:  0.1,-1 + 1) {$x$};              
  
   \addplot[color=black,mark=*,mark options={fill=white}]
   coordinates{
   
        (0.1,-1 +1)  };  \addplot[color=gray,mark=*]
   coordinates{
   
        (0.1,0.29)  };  
        
        \addplot[color=black,mark=*]
   coordinates{
   
        (0.1,-0.31)  };   \end{axis}
\end{tikzpicture}
\caption{Voter's payoff realization as a function of first-period policy. The thick (thin) curves represent the case in which $x_V=\overline{\alpha}$ ($x_V=\underline{\alpha}$). Solid curves are the voter's expected payoff $E[U^V_1]$, dashed ones represent $E[U^V_1]-\frac 1{2\psi}$ and $E[U^V_1]+\frac 1{2\psi}$}
\end{center}
\end{figure}
\end{document} 